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July  2021, 17(4): 2243-2264. doi: 10.3934/jimo.2020067

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

1. 

School of Mathematical Science, Changsha Normal University, Changsha 410100, Hunan, China

2. 

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author: Jinbiao Wu

Received  August 2019 Revised  January 2020 Published  March 2020

Fund Project: The first author is supported by Provincial Natural Science Foundation of Hunan under Grant 2019JJ50677 and the Program of Hehua Excellent Young Talents of Changsha Normal University

We consider an $ M^X/G/1 $ retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until $ N $ or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.

Citation: Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2243-2264. doi: 10.3934/jimo.2020067
References:
[1]

J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (20100), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar

[2]

J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211.  doi: 10.1007/BF02564721.  Google Scholar

[3]

A. Aissani, On the M/G/1 queueing system with repeated orders and unreliable server, Journal of Technology, 6 (1988), 93-123.   Google Scholar

[4]

E. Altman and U. Yechiali, Analysis of customers'impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279.  doi: 10.1007/s11134-006-6134-x.  Google Scholar

[5]

F. M. ChangT. H. Liu and J. C. Ke, On an unreliable-server retrial queue with customer feedback and impatience, Applied Mathematical Modelling, 55 (2018), 171-182.  doi: 10.1016/j.apm.2017.10.025.  Google Scholar

[6]

J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Quality Technology & Quantitative Management, 15 (2018), 589-601.  doi: 10.1080/16843703.2017.1320459.  Google Scholar

[7]

G. ChoudhuryJ. Ke and L. Tadj, The $N$-policy for an unreliable server with delaying repair and two phases of service, Journal of Computational and Applied Mathematics, 231 (2009), 349-364.  doi: 10.1016/j.cam.2009.02.101.  Google Scholar

[8]

R. B. Cooper, Introduction to Queueing theory, North-Holland, New York, 1981.  Google Scholar

[9]

G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-167.  doi: 10.1007/BF01158472.  Google Scholar

[10]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. Google Scholar

[11]

G. Jain and K. Sigman, A Pollaczek-Khintchine formula for M/G/1 queues with disasters, Journal of Applied Probability, 33 (1996), 1191-1200.  doi: 10.2307/3214996.  Google Scholar

[12]

V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems, 7 (1990), 191-208.  doi: 10.1007/BF01158474.  Google Scholar

[13]

Q. L. LiY. DuG. Dai and M. Wang, On a doubly dynamically controlled supermarket model with impatient customers, Computers & Operations Research, 55 (2015), 76-87.  doi: 10.1016/j.cor.2014.10.004.  Google Scholar

[14]

H. S. Lee and M. M. Srinivasan, Control policies for $M^X/G/1$ queueing system, Management Science, 35 (1989), 708-721.  doi: 10.1287/mnsc.35.6.708.  Google Scholar

[15]

H. W. LeeS. S. Lee and K. C. Chae, Operating characteristics of $M^X/G/1$ queue with $N$-policy, Queueing Systems, 15 (1994), 387-399.  doi: 10.1007/BF01189247.  Google Scholar

[16]

S. S. LeeH. W. LeeS. H. Yoon and K. C. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189.   Google Scholar

[17]

Z. LiuY. Chu and J. Wu, Heavy-traffic asymptotics of a priority polling system with threshold service policy, Computers & Operations Research, 65 (2016), 19-28.  doi: 10.1016/j.cor.2015.06.013.  Google Scholar

[18]

G. C. Mytalas and M. A. Zazanis, An $M^X/G/1$ queueing system with disasters and repairs under a multiple adapted vacation policy, Naval Research Logistics, 62 (2015), 171-189.  doi: 10.1002/nav.21621.  Google Scholar

[19]

T. Phung-Duc, Multiserver retrial queues with two types of nonpersistent customers, Asia-Pacific Journal of Operational Research, 31 (2014), 1440009, 27pp. doi: 10.1142/S0217595914400090.  Google Scholar

[20]

N. Perel and U. Yechiali, Queues with slow servers and impatient customers, European Journal of Operational Research, 201 (2010), 247-258.  doi: 10.1016/j.ejor.2009.02.024.  Google Scholar

[21]

A. G. Pakes, Some conditions for ergodicity and recurrence of Markov chains, Operations Research, 17 (1969), 1058-1061.  doi: 10.1287/opre.17.6.1058.  Google Scholar

[22]

T. Phung-Duc, Retrial queueing models: A survey on theory and applications, preprint, arXiv: 1906.09560, 2019. Google Scholar

[23]

L. I. SennotP. A. Humblet and R. L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 785-789.  doi: 10.1287/opre.31.4.783.  Google Scholar

[24]

J. WangJ. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380.  doi: 10.1023/A:1010918926884.  Google Scholar

[25]

J. WangB. Liu and J. Li, Transient analysis of an M/G/1 retrial queue subject to disasters and server failures, European Journal of Operational Research, 189 (2008), 1118-1132.  doi: 10.1016/j.ejor.2007.04.054.  Google Scholar

[26]

J. Wu and Z. Lian, A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule, Computers & Industrial Engineering, 64 (2013), 84-93.  doi: 10.1016/j.cie.2012.08.015.  Google Scholar

[27]

R. W. Wolff, Poisson arrival see time average, Operations Research, 30 (1982), 223-231.  doi: 10.1287/opre.30.2.223.  Google Scholar

[28]

T. YangM. J. M. Posner and J. G. C. Templeton, The M/G/1 retrial queue with nonpersistent customers, Queueing Systems, 7 (1990), 209-218.  doi: 10.1007/BF01158475.  Google Scholar

[29]

T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems, 2 (1987), 201–233. doi: 10.1007/BF01158899.  Google Scholar

[30]

U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202.  doi: 10.1007/s11134-007-9031-z.  Google Scholar

show all references

References:
[1]

J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (20100), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar

[2]

J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211.  doi: 10.1007/BF02564721.  Google Scholar

[3]

A. Aissani, On the M/G/1 queueing system with repeated orders and unreliable server, Journal of Technology, 6 (1988), 93-123.   Google Scholar

[4]

E. Altman and U. Yechiali, Analysis of customers'impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279.  doi: 10.1007/s11134-006-6134-x.  Google Scholar

[5]

F. M. ChangT. H. Liu and J. C. Ke, On an unreliable-server retrial queue with customer feedback and impatience, Applied Mathematical Modelling, 55 (2018), 171-182.  doi: 10.1016/j.apm.2017.10.025.  Google Scholar

[6]

J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Quality Technology & Quantitative Management, 15 (2018), 589-601.  doi: 10.1080/16843703.2017.1320459.  Google Scholar

[7]

G. ChoudhuryJ. Ke and L. Tadj, The $N$-policy for an unreliable server with delaying repair and two phases of service, Journal of Computational and Applied Mathematics, 231 (2009), 349-364.  doi: 10.1016/j.cam.2009.02.101.  Google Scholar

[8]

R. B. Cooper, Introduction to Queueing theory, North-Holland, New York, 1981.  Google Scholar

[9]

G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-167.  doi: 10.1007/BF01158472.  Google Scholar

[10]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. Google Scholar

[11]

G. Jain and K. Sigman, A Pollaczek-Khintchine formula for M/G/1 queues with disasters, Journal of Applied Probability, 33 (1996), 1191-1200.  doi: 10.2307/3214996.  Google Scholar

[12]

V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems, 7 (1990), 191-208.  doi: 10.1007/BF01158474.  Google Scholar

[13]

Q. L. LiY. DuG. Dai and M. Wang, On a doubly dynamically controlled supermarket model with impatient customers, Computers & Operations Research, 55 (2015), 76-87.  doi: 10.1016/j.cor.2014.10.004.  Google Scholar

[14]

H. S. Lee and M. M. Srinivasan, Control policies for $M^X/G/1$ queueing system, Management Science, 35 (1989), 708-721.  doi: 10.1287/mnsc.35.6.708.  Google Scholar

[15]

H. W. LeeS. S. Lee and K. C. Chae, Operating characteristics of $M^X/G/1$ queue with $N$-policy, Queueing Systems, 15 (1994), 387-399.  doi: 10.1007/BF01189247.  Google Scholar

[16]

S. S. LeeH. W. LeeS. H. Yoon and K. C. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189.   Google Scholar

[17]

Z. LiuY. Chu and J. Wu, Heavy-traffic asymptotics of a priority polling system with threshold service policy, Computers & Operations Research, 65 (2016), 19-28.  doi: 10.1016/j.cor.2015.06.013.  Google Scholar

[18]

G. C. Mytalas and M. A. Zazanis, An $M^X/G/1$ queueing system with disasters and repairs under a multiple adapted vacation policy, Naval Research Logistics, 62 (2015), 171-189.  doi: 10.1002/nav.21621.  Google Scholar

[19]

T. Phung-Duc, Multiserver retrial queues with two types of nonpersistent customers, Asia-Pacific Journal of Operational Research, 31 (2014), 1440009, 27pp. doi: 10.1142/S0217595914400090.  Google Scholar

[20]

N. Perel and U. Yechiali, Queues with slow servers and impatient customers, European Journal of Operational Research, 201 (2010), 247-258.  doi: 10.1016/j.ejor.2009.02.024.  Google Scholar

[21]

A. G. Pakes, Some conditions for ergodicity and recurrence of Markov chains, Operations Research, 17 (1969), 1058-1061.  doi: 10.1287/opre.17.6.1058.  Google Scholar

[22]

T. Phung-Duc, Retrial queueing models: A survey on theory and applications, preprint, arXiv: 1906.09560, 2019. Google Scholar

[23]

L. I. SennotP. A. Humblet and R. L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 785-789.  doi: 10.1287/opre.31.4.783.  Google Scholar

[24]

J. WangJ. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380.  doi: 10.1023/A:1010918926884.  Google Scholar

[25]

J. WangB. Liu and J. Li, Transient analysis of an M/G/1 retrial queue subject to disasters and server failures, European Journal of Operational Research, 189 (2008), 1118-1132.  doi: 10.1016/j.ejor.2007.04.054.  Google Scholar

[26]

J. Wu and Z. Lian, A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule, Computers & Industrial Engineering, 64 (2013), 84-93.  doi: 10.1016/j.cie.2012.08.015.  Google Scholar

[27]

R. W. Wolff, Poisson arrival see time average, Operations Research, 30 (1982), 223-231.  doi: 10.1287/opre.30.2.223.  Google Scholar

[28]

T. YangM. J. M. Posner and J. G. C. Templeton, The M/G/1 retrial queue with nonpersistent customers, Queueing Systems, 7 (1990), 209-218.  doi: 10.1007/BF01158475.  Google Scholar

[29]

T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems, 2 (1987), 201–233. doi: 10.1007/BF01158899.  Google Scholar

[30]

U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202.  doi: 10.1007/s11134-007-9031-z.  Google Scholar

Figure 1.  The mean system size versus $ p $ with $ (\alpha, \delta, \theta) = (0.1, 3, 0.7) $
Figure 2.  The mean system size versus $ \delta $ with $ (\alpha, p, \theta) = (0.1, 0.1, 0.7) $
Figure 3.  The mean system size versus $ \theta $ with $ (\alpha, p, \delta) = (0.1, 0.1, 3) $
Figure 4.  The mean system size versus $ \alpha $ with $ (\theta, p, \delta) = (0.7, 0.1, 3) $
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